9.1 – Inverse and Joint Variation Inverse Variation – Two variables x and y show inverse variation provided y = k/x. * Direct Variation – Two variables x and y show direct variation provided y = kx. * Constant of Variation – The nonzero constant (usually donated k) in direct variation equation, an inverse variation equation, or a joint variation equation. Joint Variation – A relationship that occurs when a quantity varies directly as the product of two or more other quantities: z = kxy. *
*when k is a nonzero constant.
9.2 – Graphing Simple Rational Fractions Rational Function – A function of the form, f(x) = p(x) ÷ q(x), where p(x) and q(x) are polynomials and q(x) ≠ 0. Hyperbola – The set of all points P such that the difference of the distance from P to two fixed points, called foci, is constant. Branches – Two symmetrical parts of a hyperbola.
9.3 – Graphing General Rational Functions Characteristics of Graphs of Rational Functions:f(x) = p(x) ÷ q(x)
1) The x-intercepts of the graph of f are the real zeros of p(x).
2) The graph of f has a vertical asymptote at each real zero of q(x).
3) The graph of f has at most one horizontal asymptote.
- If m < n, the line y = 0 is a horizontal asymptote
- If m = n, the line y = am ÷ bn is a horizontal asymptote.
- If m > n, the graph has no horizontal asymptote.
9.4 – Multiplying and Dividing Rational Expressions Simplified Form – A rational expression in which the numerator and denominator have no common factors (other than ±1).
- Let a, b, and c be nonzero real numbers or variable expressions. Then the following rule applies: ac = a
bc = b
9.5 – Addition, Subtraction, and Complex Fractions How to add or subtract fractions with like denominators:
- Simply add/subtract the numerators and place the result over the denominator. How to add or subtract fractions with unlike denominators:
1) Find the LCD
2) Rewrite each expression as an equivalent expression using the LCD.
3) Add/ Subtract as if they were rational expression with like denominators.
4) Simplify, if necessary. Complex Fraction – A fraction that contains a fraction in it numerator or denominator. How to Simplify Complex Fractions:
1) Add fractions in numerator and denominator separately (Follow rules above).
2) Multiply by denominator reciprocal.
3) Divide out common factors and simplify.
9.6 – Solving Rational Equations How to solve Rational Equations in which there is more than one rational expression on a given side:
1) Find the LCD
2) Multiply both sides of the equation by the LCD.
3) Simplify and solve the polynomial equation left.
4) Check for extraneous solutions. Cross Multiplying - A method of solving a simple rational equation for which each side of the equation is a single rational expression. (Multiply one numerator by the other fraction’s denominator, solve, and don’t forget to check for extraneous solutions.)
9.4 Vid Vid on multiplyig an dviding rational expressions
Practice Problems 9.1
1.) x and y vary inversely, use the given values to write an equation relating x and y. Then find y when x=2.
x=5, y=-2
(5)(-2)=k
k=-10
y=-10/x
y=-10/2
y=-5
2.) z varies jointly with x and y, use the given values to write an equation relating x,y,and z. Then find z when x=-4 and y=7.
x=8, y=5, z=4
4=(8)(5)(k)
k=1/10
z=1/10xy
z=(1/10)(4)(7)
z= 2.8
9.2
1.)Identify the horizontal and vertical asymptotes of the graph of the function. Then state the domain and range.
y=3/x+2
y=2 and x=0
domain= all real numbers, except 0
range= all real numbers, except 2
9.3
1.) Identify the x-intercepts and vertical asymptotes of the graph of the function.
y=x^2+4x-5/x-6
x-intercepts:-5,1
vertical asymptote x=6
9.4
1.) Simplify the rational expression.
x^3-27/x^3+3x^2+9x
x-3/x
2.)Multiply the rational expression.
80x^4/y^3 * xy/5x^2
=16x^3/y^2
9.5
1.) Perform the indicated operation.
23/10x^2 - x/10x^2
=23-x/10x^2
2.)Find the least common denominator.
4/21x^2, x/3x^2-15x
21x^2(x-5)
3.)Perform the indicated operation.
-4/7x - 5/3x
=-47/21x
9.6
1.)Solve the equation by using the LCD.
3/2+1/x=2
=2
2.)Solve the equation by cross multiplying.
8(x-1)/x^2-4=4/x-2
=4
3.)Solve the equation using any method.
3/x+2=6/x-1
=-5
4.)x/2x-6=2/x-4
=2,6
The concepts learned in chapter nine can be applied to real life. Variations can be applied to things affecting other things, such as in direct, indirect and joint variations. One example would be the amount of force from an object and its effect on a brick wall. Maybe the effect on the wall from the force of someone’s head versus that of a car. Writing and simplifying rational expressions is used when one is given a large number of measurements combined with variables, such as the surface area of a person, if you would care to know the surface area your skin covers when laid out flat. As there seems to be no careers directly related to the sections covered in this chapter, I am unable to give any examples of one.
Definitions :)
9.1 – Inverse and Joint VariationInverse Variation – Two variables x and y show inverse variation provided y = k/x. *
Direct Variation – Two variables x and y show direct variation provided y = kx. *
Constant of Variation – The nonzero constant (usually donated k) in direct variation equation, an inverse variation equation, or a joint variation equation.
Joint Variation – A relationship that occurs when a quantity varies directly as the product of two or more other quantities: z = kxy. *
*when k is a nonzero constant.
9.2 – Graphing Simple Rational Fractions
Rational Function – A function of the form, f(x) = p(x) ÷ q(x), where p(x) and q(x) are polynomials and q(x) ≠ 0.
Hyperbola – The set of all points P such that the difference of the distance from P to two fixed points, called foci, is constant.
Branches – Two symmetrical parts of a hyperbola.
9.3 – Graphing General Rational Functions
Characteristics of Graphs of Rational Functions: f(x) = p(x) ÷ q(x)
1) The x-intercepts of the graph of f are the real zeros of p(x).
2) The graph of f has a vertical asymptote at each real zero of q(x).
3) The graph of f has at most one horizontal asymptote.
- If m < n, the line y = 0 is a horizontal asymptote
- If m = n, the line y = am ÷ bn is a horizontal asymptote.
- If m > n, the graph has no horizontal asymptote.
9.4 – Multiplying and Dividing Rational Expressions
Simplified Form – A rational expression in which the numerator and denominator have no common factors (other than ±1).
- Let a, b, and c be nonzero real numbers or variable expressions. Then the following rule applies:
ac = a
bc = b
9.5 – Addition, Subtraction, and Complex Fractions
How to add or subtract fractions with like denominators:
- Simply add/subtract the numerators and place the result over the denominator.
How to add or subtract fractions with unlike denominators:
1) Find the LCD
2) Rewrite each expression as an equivalent expression using the LCD.
3) Add/ Subtract as if they were rational expression with like denominators.
4) Simplify, if necessary.
Complex Fraction – A fraction that contains a fraction in it numerator or denominator.
How to Simplify Complex Fractions:
1) Add fractions in numerator and denominator separately (Follow rules above).
2) Multiply by denominator reciprocal.
3) Divide out common factors and simplify.
9.6 – Solving Rational Equations
How to solve Rational Equations in which there is more than one rational expression on a given side:
1) Find the LCD
2) Multiply both sides of the equation by the LCD.
3) Simplify and solve the polynomial equation left.
4) Check for extraneous solutions.
Cross Multiplying - A method of solving a simple rational equation for which each side of the equation is a single rational expression. (Multiply one numerator by the other fraction’s denominator, solve, and don’t forget to check for extraneous solutions.)
9.4 Vid Vid on multiplyig an dviding rational expressions
Practice Problems
9.1
1.) x and y vary inversely, use the given values to write an equation relating x and y. Then find y when x=2.
x=5, y=-2
(5)(-2)=k
k=-10
y=-10/x
y=-10/2
y=-5
2.) z varies jointly with x and y, use the given values to write an equation relating x,y,and z. Then find z when x=-4 and y=7.
x=8, y=5, z=4
4=(8)(5)(k)
k=1/10
z=1/10xy
z=(1/10)(4)(7)
z= 2.8
9.2
1.)Identify the horizontal and vertical asymptotes of the graph of the function. Then state the domain and range.
y=3/x+2
y=2 and x=0
domain= all real numbers, except 0
range= all real numbers, except 2
9.3
1.) Identify the x-intercepts and vertical asymptotes of the graph of the function.
y=x^2+4x-5/x-6
x-intercepts:-5,1
vertical asymptote x=6
9.4
1.) Simplify the rational expression.
x^3-27/x^3+3x^2+9x
x-3/x
2.)Multiply the rational expression.
80x^4/y^3 * xy/5x^2
=16x^3/y^2
9.5
1.) Perform the indicated operation.
23/10x^2 - x/10x^2
=23-x/10x^2
2.)Find the least common denominator.
4/21x^2, x/3x^2-15x
21x^2(x-5)
3.)Perform the indicated operation.
-4/7x - 5/3x
=-47/21x
9.6
1.)Solve the equation by using the LCD.
3/2+1/x=2
=2
2.)Solve the equation by cross multiplying.
8(x-1)/x^2-4=4/x-2
=4
3.)Solve the equation using any method.
3/x+2=6/x-1
=-5
4.)x/2x-6=2/x-4
=2,6
The concepts learned in chapter nine can be applied to real life. Variations can be applied to things affecting other things, such as in direct, indirect and joint variations. One example would be the amount of force from an object and its effect on a brick wall. Maybe the effect on the wall from the force of someone’s head versus that of a car. Writing and simplifying rational expressions is used when one is given a large number of measurements combined with variables, such as the surface area of a person, if you would care to know the surface area your skin covers when laid out flat. As there seems to be no careers directly related to the sections covered in this chapter, I am unable to give any examples of one.